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Theses

Formalisation en logique linéaire du fonctionnement des réseaux de Petri

Abstract : In classical logic, the formalisation of the Petri net token game faces the invariance of truth. In modal logic, it requires first to construct the reachable marking graph. In contrast, Girard's linear logic allows us to use purely propositional provable sequents to directly formalise the reachability relations in a net: each transition appears as a linear implication available ad libitum between the propositions translating its input and output markings. To study thoroughly this formalisation, we define as primitives in linear logic the notions of resource, action and consummability/producibility, analogous but distinct from those of proposition, deduction and truth/falseness in classical logic. We establish a concrete interpretation of the linear connectives with respect to their syntactical properties. The connective "par" is presented as an operator of disjoint accumulation of resource items (dual of the connective ¿times¿ of joint accumulation) and the linear negation "nil" as a time inversion. This concrete view brings out the limits of the existing formalisation of the token game. We increase the expressive power of these formalisations by translating each transitions as an ordinary linear implication, being itself a perishable resource, whose consuming represents a transition occurrence in firing sequence. We give a logical expression to the essential features of the Petri net token game : we demonstrate that a reachability relation by firing sequence is equivalent to a provable sequent and that the fundamental equation is an algebraic expression of the balancing criterion of linear logic. By means of the combinatory of all linear connectives, our approach opens perspectives for analysing complex reachability relations, e. g. for recovering after crash in an industrial system.
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https://tel.archives-ouvertes.fr/tel-00010245
Contributor : Emilie Marchand <>
Submitted on : Thursday, September 22, 2005 - 11:01:26 AM
Last modification on : Friday, January 10, 2020 - 9:08:09 PM
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  • HAL Id : tel-00010245, version 1

Citation

François Girault. Formalisation en logique linéaire du fonctionnement des réseaux de Petri. Automatique / Robotique. Université Paul Sabatier - Toulouse III, 1997. Français. ⟨tel-00010245⟩

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